Pussy Riot members Maria Alyokhina and Nadezhda Tolokonnikova are demanding the Russian government pay them compensation of over $300,000 for what they allege amounts to torture during their prosecution and subsequent imprisonment for a 2012 "punk prayer" in a Moscow cathedral.

The women, who were released last December after serving 21 months in prison and pre-trial detention are suing the Kremlin in the European Court of Human Rights.

They are demanding $160,000 each in compensation and $13,000 to cover court costs, arguing that their investigation and prosecution violated their rights and amounted to torture.

The pair say that Russia, which is a member of the European convention on human rights, violated four key articles guaranteeing the rights to freedom of expression, liberty and security and a fair trial, and prohibiting torture.

The women were tried in a court where they were held in glass or barred cages and transport from pre-trial detention to the court took up to four hours, accompanied at all times by police dog handlers.

If their bid is successful the pair plan to challenge their conviction for religiously hooliganism and seek to have it overturned in Russia.

Pavel Chikov,head of Russian human rights legal group Agora, which is representing the two, said neither woman had received a fair trial and the case could set a precedent for free speech in Russia.

The two would donate any compensation they receive to human rights groups, including their own penal reform organization.